Statistical Theory of Breakup Reactions

Carlos A. Bertulani1∗, Pierre Descouvemont2†, Mahir S. Hussein3‡

Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 75429, USAPhysique Nucléaire Théorique et Physique Mathématique, C.P. 229, Université Libre de Bruxelles (ULB), B 1050

Brussels, BelgiumInstituto de Estudos Avançados, Universidade de São Paulo C. P. 72012, 05508-970 São Paulo-SP, Brazil, and

Instituto de F́ısica, Universidade de São Paulo, C. P. 66318, 05314-970 São Paulo,-SP, Brazil

March 25, 2014

Abstract

We propose alternatives to coupled-channels calculations with loosely-bound exotic nuclei (CDCC),based on the the random matrix (RMT) and the optical background (OPM) models for the statisticaltheory of nuclear reactions. The coupled channels equations are divided into two sets. The first set,described by the CDCC, and the other set treated with RMT. The resulting theory is a StatisticalCDCC (CDCCS), able in principle to take into account many pseudo channels.

1 Statistical Continuum Discretized Coupled Channels

Continuum-Discretized Coupled-Channels (CDCC) calculations are a major theoretical tool to cal-culate observables in reactions involving rare loosely-bound nuclear isotopes [1, 2]. Such calculationsare time-consuming and may include such a huge humber of channels that they are amenable to astatistical treatment, similar to what has been used to treat neutron-induced reactions with compoundnuclear states. This is the subject of the present work.

We write the CDCC equations in a schematic model as,[− ~

2

2µ

d2

dx2+ Vrel(x) + �n − E

]ψn(x) +

∑m

Vnm(x)ψm(x) = 0 (1)

∗carlos.bertulani@tamuc.edu†pdesc@ulb.ac.be‡hussein@if.usp.br

1

arX

iv:1

403.

5820

v1 [

nucl

-th]

24

Mar

201

4

Figure 1: Strongly coupled states belonging to space P and denoted by roman letters, are weakly coupledto states belonging to space Q and denoted by greek letters.

We distinguish the desired channels in a conventional CDCC by the labels m,n, from the statisticalchannels labeled by µ, ν. Accordingly,[

− ~2

2µ

d2

dx2+ Vrel(x) + �n − E

]ψn(x) +

∑m

Vnm(x)ψm(x) +∑µ

Vnµ(x)ψµ(x) = 0 (2)

and [− ~

2

2µ

d2

dx2+ Vrel(x) + �µ − E

]ψµ(x) +

∑ν

Vµν(x)ψν(x) +∑m

Vµm(x)ψm(x) = 0 (3)

The statistical nature of the µ, ν channels is specified through the following properties of the matrixelements,

Vµν = 0 (4)

VµνVηρ = (δµ,ηδν,ρ + δµ,ρδν,η)V 2µ,ν (5)

2

where the second moment V 2µ,ν can be parametrized as,

V 2µ,ν =ω0√

ρ(�µ)ρ(�ν)exp

{−(�µ + �ν)

2

2∆2

}(6)

with ρ(�) being the density of states, and ω0 and ∆ are adjustable parameters.The same statistical properties are invoked on the matrix elements Vµ,m, etc.

Vµm = 0, (7)

VµmVηn = (δµ,ηδ(m,n) + δµ,nδn,η)V 2µ,m, (8)

where the second moment V 2µ,m can be parametrized as,

V 2µ,m =ω0ρ(�µ)

exp

{−(�µ + �η)

2

2∆2m

}(9)

with ρ(�) being the density of states, and ω0 and ∆ are adjustable parameters.

The above prescription should set the stage for a CDCC calculation which presents fluctuationsin the final channels and one must rely on an appropriate ensemble average: perform the calculationseveral times and at the end perform an average.

2 CDCCS in the random matrix formalism

It is natural to expect that the CDCC equations for the m,n channels would be affected by thestatistical channels (µ, ν). Averaging the equations above is rather difficult. An easier procedure is toeliminate the statistical channels in favor of the CDCC channels (m,n), resulting in

ψµ(x) =

∫ ∞0

dx′Gµ(x, x′)∑m

Vµm(x′)ψm(x

′) (10)

where Gµ(x, x′) is the diagonal elements of a matrix Green’s function defined through the equation[

[− ~2

2µ

d2

dx2+ Vrel(x) + �µ − E]δµ,ν + Vµν(x)

]Gµ,ν(x, x

′) = δ(x− x′) (11)

With the above Green’s function we have for the effective CDCC equations,[− ~

2

2µ

d2

dx2+ Vrel(x) + Vpol,n(x) + �n − E

]ψn(x) +

∑m

Vnm(x)ψm(x) = 0 (12)

where Vpol,n(x), is given by

Vpol,n(x)ψn(x) =

∫ ∞0

dx∑µ

Vnµ(x)

∫ ∞0

dr′Gµ(x, x′)∑m

Vµm(x′)ψm(x

′) (13)

3

The polarization potential fluctuates owing to the random nature of the coupling Vn,µ. Thus wehave to average this equation over the ensemble. What remains is the fluctuation contribution whichwe address later. Thus,

Vpol,n(x)ψn(x) =∑µ

Vnµ(x)

∫ ∞0

dx′Gµ(x, x′)∑m

Vµm(x′)ψm(x′). (14)

The above equation is difficult to average. What we can do is to borrow from the Optical Back-ground Representation of KKM, and introduce the average polarization potential,

Vpol,n(x) =∑µ

Vnµ(x)

∫ ∞0

dx′Gµ(x, x′)∑m

Vµm(x′) (15)

To perform the average above, we have to expand the Green’s function in Vµν and then consideronly even powers of V,s. This is quite lengthy and was done by Weidenmüller. Here we ignorethe fluctuations in G, and proceed to average Vnµ(x)Vµm(x′) = [δn,µδµ,m + δn,m]F (x, x

′)V 2µ,m, whereF (x, x′) = exp

{−(x− x′)2/2σ2

}.

Accordingly, we have the CDCCS equations with the average polarization potential read,[− ~

2

2µ

d2

dx2+ Vrel(x) + Vpol,n(x) + �n − E

]ψn(x) +

∑m

Vnm(x)ψm(x) = 0. (16)

The above equations are the new CDCC ones appropriate for our purpose. The fluctuation contri-bution can be obtained by going back to the original equation and write Vpol,n(x) = Vpol,n(x)+V

flpol,n(x).

This implies that the CDCC wave functions themselves fluctuate. However, the CDCC equations arenot the appropriate venue to obtain the fluctuation contribution. What we need is an equation for thesquare of the CDCC wave functions: a master equation. This was obtained by Weidenmülcer [4], and[5]. We should mention that fluctuations in the final state has been discussed in the case of transferleading to the excitation of an isobaric analog resonance [6].

3 Time-dependent CDCCS

In the c.m. of the projectile, we assume H = H0 + V , where H is composed by the non-perturbedHamiltonian H0 and a small perturbation V . The Hamiltonian H0 satisfies an eigenvalue equationH0ψn = Enψn, whose eigenfunctions form a complete basis (including continuum) in which the totalwavefunction Ψ, that obeys HΨ = i~∂Ψ/∂t, can be expanded as Ψ =

∑n an(t)ψne

−iEnt/~. One obtains

i~∑n

ȧnψne−iEnt/~ =

∑n

V anψne−iEnt/~, (17)

with ȧn ≡ dan(t)/dt. Using the orthogonalization properties of the ψn, let us multiply (17) by ψ∗kand integrate it in the coordinate space. From this, results the the time-dependent coupled channelsequations

ȧk (t) = −i

~∑n

an (t)Vkn (t) eiEk−En

~ t, (18)

4

Figure 2: Scematic representation of the P and Q spaces in the optical background formalism.

where we introduced the matrix element (dτ is the volume element) Vkn =∫ψ∗kV ψn dτ.

In order to get balance equations, we follow a method described in Ref. [7] (section 3.2.3) for theexcitation of giant resonances in relativistic heavy Ion collisions. We rewrite Eq. (18) with explicitaccount of P (bound + “strongly-coupled states” in continuum - denoted by roman letters) + Q(“weakly-coupled” states in the continuum - denoted by greek letters)

ȧk (t) = −i

~∑n

an (t)Vkn (t) eiEk−En

~ t − i~∑n

aµ (t)Vkµ (t) eiEk−Eµ

~ t +

∫ ∞−∞

dt′Kk(t− t′)ak(t′), (19)

where we introduced the last term to account for decay to other channels than the breakup channelunder scrutiny. This function is given in terms of the width of the state k as

K(t− t′) = − i4π

∫ ∞−∞

dω eiω(t−t′) Γk

(ω − Ek

~

). (20)

For Γk equal to a constant, one obtains Kk(t− t′) = −i(Γk/2)δ(t− t′), we get

ȧk (t) = −i

~∑n

an (t)Vkn (t) eiEk−En

~ t − i~∑n

aµ (t)Vkµ (t) eiEk−Eµ

~ t − iΓk2ak(t). (21)

Since Γk 6= 0, the total probability P =∑

k |ak(t)|2 is no longer conserved because a flux is nowput into the decay channels. Multiplying Eq. (21) by a∗k and its complex conjugate by ak(t) andsubtracting the results, we obtain for the occupation probability Pk(t) = |ak(t)|2,

Ṗk (t) =2

~=

[∑n

a∗n(t)ak(t)Vkn (t) eiEk−En

~ t

]+

2

~=

[∑n

a∗k(t)aµ (t)Vkµ (t) eiEk−Eµ

~ t

]− Γk

~Pk(t). (22)

This equation can be written as a balance equation for the probability in the form

Ṗk (t) = Gk(t) + Lk(t), (23)

where the gain term is obtained whenever

Hk(t) =2

~=

[∑n

a∗n(t)ak(t)Vkn (t) eiEk−En

~ t

]+

2

~=

[∑n

a∗k(t)aµ (t)Vkµ (t) eiEk−Eµ

~ t

], (24)

5

is positive, i.e., G(+)k = positive(Hk), and the loss term is obtained whenever Hk(t) is negative added

to the loss by decay,

Lk (t) = G(−)k (t)−

Γk~Pk(t). (25)

Similar equations are obtained for the occupation probability Pµ due to nµ (PQ) and νµ (QQ)coupling. The first part on the rhs of Eq. (24) is obtained directly from solving the cc equationswith the “strongly interacting” P states. But the second term has to be averaged in a proper way.The same needs to be done for both terms of Hµ which also contains terms proportional to aµaν . Ifthe averaging is possible then the cc equation can be reiterated to include the couplings involving thecontinuum till convergence is achieved.

4 Optical background CDCCS

We consider the scattering of a projectile with center of mass energy E, which by means of theinteraction with the target and between the core and neutron (say, for 11Be → 10Be+n) may transitto other channels. We consider scattering within a narrow band of discretized continuum statesbelonging to a space P. All other states, continuum + bound states, will be assumed to belong tospace Q. We emphasize here that this decomposition of the Hilbert space is done on the final wavefunction which contains the breakup channels. As such the energy mentioned above should be takenas that pertaining to the nucleus which breaks up. If the energy is taken as the total CM one and thedecomposition is done on the total wave function, then we are dealing with the conventional compoundnucleus reaction, which is not the aim of this investigation. This is an extension to breakup reactionsof a formalism developed in Ref. [8]. In the following we take E to be the energy of the subsystem.

The Schrödinger equation (SE) for this problem is HΨ = EΨ, containing an internal (core-neutron)interaction V and the projectile interaction with the target U . Using the Feshbach formalism, weintroduce the projection operators P and Q, so that Q = 1 − P and Ψ = PΨ + QΨ. Then, for thepart of the wavefunction in space P , we get

(E −HPP −HPQGQHQP )PΨ = 0, (26)

where

GQ ≡ GQ (E) =1

E −HQQ + iε. (27)

The continuum is discretized by averaging the wavefunction in the channel P is over an energyinterval ∆ according to the prescription

〈Ψ〉∆ =∆

2π

∫ ∞−∞

dE′ΨE′

(E − E′)2 + ∆2/4= Ψ

(E + i

∆

2

), (28)

where the residue theorem was used.It is then straightforward to show that

PΨ = Ψ∆P + GPVPQ1

E −HQQ − VQPGPVPQVQPΨ∆P , (29)

6

Figure 3: Ratio between the average of the fluctuating part and the optical part of the T-matrix.

with Ψ∆P = 〈PΨ〉∆ ' PΨ(E + i∆/2) being the solution of the energy average of Eq. (26), which isobtained by replacing GQ(E) by GQ(E + i∆/2). This follows from the assumption that solutions ofHPP and HPQ are slowly varying functions of E. In Eq. (29),

GP (E) =1

E −HPP −HPQGQ (E + i∆/2)HQP, (30)

and

VPQ (E) = HPQ (E)(

i∆/2

E + i∆/2−HQQ + iε

)1/2VQP (E) =

(i∆/2

E + i∆/2−HQQ + iε

)1/2HQP (E) . (31)

We now consider the Hamiltonian

W = HQQ + VQPGoptP VPQ (32)

7

which describes the coupling of the Q and P spaces, e.g. bound-state-continuum and continuum-continuum couplings. We assume that a complete set of projectile eigenstates |ω〉, with eigenenergiesEω can be found for W . The eigenvalue problems are very different: ΨoptP is solution of HPP −HPQGQ

(E + i∆2

)HQP , whereas |ω〉 is solution of HQQ + VQPGoptP VPQ.

The S-matrix, Scc′ =〈

(PΨ)(−)c | (PΨ)(+)c′

〉can be obtained using steps similar to the derivation of

the Gell-Mann-Goldberger relation (or two-potential formula). One gets

Scc′ (E) = Scc′ (E)− i∑ω

γωcγωc′

E − Eω, (33)

whereγωc(E) =

√2π〈ω |VQP (E)|Ψ∆P=c

〉. (34)

The channels c and c′ in eq. (33) denote the different channels for the scattering matrix Scc′ .

The result above splits the S-matrix into a direct part, Scc′ (E) =〈

Ψ∆(−)c | (Ψ)∆(+)c′

〉, and a multi-

step part, the second term in Eq. (33), which contains all the multiple couplings between the Q andP spaces. Assuming that γωc are smooth functions of E, we can perform the ensemble average of thecoupling term in Eq. (33) as〈∑

ω

γωcγωc′

E − Eω

〉∆

=∑ω

γωc(E + i∆/2)γωc′(E + i∆/2)

E + i∆/2− Eω. (35)

In the channel P, we define an continuum-discretized model wave function

ΨP = ΨoptP = 〈PΨ〉∆ ' PΨE+i∆/2, (36)

where ΨoptP is the solution of[E −HPP −HPQGQ

(E + i

∆

2

)HQP

]ΨoptP = 0, (37)

or(E −HoptP

)ΨoptP = 0, with

HoptP (E) = HPP +HPQGQ(E + i

∆

2

)HQP . (38)

Eq. (37) follows from an average of eq. (26), with the assumption that solutions of HPP and HPQare slowly varying functions of E.

Using the simple relation

GQ(E)−GQ(E + i

∆

2

)= GQ(E)

i∆

2GQ

(E + i

∆

2

), (39)

and from eq. (26), by adding and subtracting HoptP , one gets[E −HoptP − VPQ (E)GQ(E)VQP (E)

]PΨ = 0, (40)

8

where VPQ and VQP are given by Eqs. (31).Now we solve eq. (40) and get

PΨ = ΨoptP + GoptP VPQ

1

E −HQQ − VQPGoptP VPQVQPΨoptP , (41)

with

GoptP (E) =1

E −HoptP (E)=

1

E −HPP −HPQGQ(E + i∆2

)HQP

. (42)

The HamiltonianW = HQQ + VQPGoptP VPQ (43)

describes the coupling of the Q and P spaces, e.g. bound-state-continuum and continuum-continuumcouplings and is the basis of the optical background formulation of the CDCCS .

Let us assume that a complete set of projectile eigenstates |q〉 can be found for W . That is[HQQ + VQPGoptP VPQ

]|q〉 = Eq |q〉 , (44)

then we can insert this set into eq. (41) and it becomes

PΨ = ΨoptP + GoptP VPQ |q〉

1

E − Eq〈q| VQP

∣∣∣ΨoptP 〉 . (45)The S-matrix, Scc′ =

〈(PΨ)(−)c | (PΨ)

(+)c′

〉can be obtained from the equation above and its com-

plex conjugate, using steps similar to the derivation of the Gell-Mann-Goldberger relation (or two-potential formula). The Hamiltonians are very different: ΨoptP is solution ofHPP−HPQGQ

(E + i∆2

)HQP ,

whereas |q〉 is solution of HQQ + VQPGoptP VPQ. One gets

Scc′ (E) = Scc′ (E)− i∑q

gqcgqc′

E − Eq, (46)

wheregqc(E) =

√2π〈q |VQP (E)|ΨoptP=c

〉. (47)

We first consider the eigenvalues of the operator

W0 = HQQ −HQPG0HPQ, (48)

where G0 is a free projectile Green’s function

G0 =1

E −HPP, (49)

and where a single particle projection operator P in spatial representation is

P =∑c

∫r2dr|r, c〉〈r, c| ≡ 1−Q (50)

9

where, for the time being, |r, c〉 is to be distinguished from the initial (or final) free projectile wave-function |φi〉 (or |φf 〉)

(Ei −HPP )|φi〉 = 0, (51)

with 〈r, c|r′, c′

〉=δ (r − r′)

rr′δcc′ . (52)

Eq. (48) can be written with intrinsic nuclear state indices displayed explicitly as

Wjk ≡ 〈Qj |W0|Qk〉 = δjkE(Q)j +HjPG

0HPk, (53)

where the second term above can be expanded by virtue of operator P in Eq. (50) as

HjPG0HPk =

∑cc′

∫r2dr

∫r′

2dr′Hjc(r)G

0cc′(r, r

′)Hc′k(r′) (54)

where Hcj(r) isHcj(r) = 〈r, c|H|Qj〉. (55)

It is now convenient to define the matrix

Mjk (Ec) = HjPG0 (Ec)HPk, (56)

in terms of which, eq. (53) can be cast into the form

Wjk = δjkE(Q)j +Mjk. (57)

Matrix Mjk can be conveniently separated into its principal value and an imaginary part

Mjk (E) = HjPP

E −HPPHPk − iπHjP δ(E −HPP )HPk ≡ Djk(E)− iΓjk(E)/2, (58)

where matrices D and Γ are defined for convenience, in a notation that alludes to resonance shifts andwidths that will be obtained by diagonalizing Wjk. A completeness relation,

∑c

∫dE|χ(+)E;c〉〈χ

(+)

E;c| = 1,for eigenstates of HPP can be used to write Γjk as

Γjk (E) = 2π∑c

HjP |χ(+)

E;c〉〈χ(+)

E;c|HPk,= 2π∑c

∫rr′Hjc(r)χ

(+)

E;c(r)χ(+)∗E;c (r

′)Hck(r′),

where for a free particle 〈r; c|χ(+)E;c′〉 ≡ χ(+)

E;cc′(r) = δcc′χ(+)

E;c(r).A dispersion relation between the real and imaginary part of Mjk in Eq. (58) yields

Djk(E) =1

2πP∫ ∞

0

Γjk(E′)dE′

E − E′. (59)

The energy dependence of Djk(E) will come from Γjk (E) and from asymmetric limits of integration.Computation of Green’s function matrix is thus simplified as it depends on (approximately) free-particle eigenfunctions alone.

10

Figure 4: Number of “counts” for the ratio between the average of the fluctuating part and the opticalpart of the T-matrix.

We write

Hcj(r) =∑k

hcjkδ(r − rk)rrk

, (60)

where hcjk are complex numbers obtained from the interaction at projectile position rk, sandwichedbetween states j and c. That is, the label k in the couplings hcjk denote the location of the spatialcoordinate where the interaction (60) assumes a non-zero value. The label c, denotes the channelindex. As we are not including any other quantum number than the energy of the channel, thediscretization of energy is the same as the label c. That is, c labels energies varying within the aninterval of width ∆ (the same energy interval used in integral (28)). The label j denotes the Q-space.In our case, it will be attached to the energy, EQj ≡ Ej . The matrix elements between channels c andc′ will not be needed. These are included in HPP and are part of the T-matrix we are after.

We now define the T-matrix Tcc′ as

Tif = T0if + 〈φi|HPQ

1

E −HQQ −HQPG0HPQHQP |φf 〉, (61)

and we diagonalize the denominator of Eq. (61) in terms of the eigenfunctions |q̂〉 of W0,

(Êq −W0)|q̂〉 = 0, (62)

11

and use that the |q̂〉’s are linear combinations of eigenstates of HQQ, (E(Q)j −HQQ)|Qj〉 = 0

|q̂〉 =∑j

Ĉjq|Qj〉. (63)

Then we can rewrite eq. (61) as1

Tif = T0if +

∑qjk

ĈjqĤij1

E − ÊqĈkq Ĥkf , (64)

where we defined

Ĥij ≡ 〈φi|PH|Qj〉 =∑c

∫r2drφ∗ic(r)〈r, c|H|Qj〉 =

∑c

∫r2drφ∗ic(r)Hcj(r). (65)

We now derive an analogous expression for the T -matrix, with ΨP defined in eq. (36). (To simplifynotations we use here G ≡ Gopt):

Tif = T if + 〈Ψi|VPQ1

E −HQQ − VQPGVPQVQP |Ψf 〉, (66)

by using the following expression for the optical Green’s function G and the continuum discretized ket|Ψc〉:

G = G0 +G0HPQ1

E −HQQ −HQPG0HPQ + i∆/2HQPG

0|Ψf 〉

= |φf 〉+G0HPQ1

E −HQQ −HQPG0HPQ + i∆/2HQP |φf 〉. (67)

Instead of diagonalizing W0 as above, we will diagonalize W , eq. (43),

W = HQQ + VQPGVPQ (68)

or in matrix notationWjk = δjkE

(Q)j + VjPGVPk (69)

where the second term can be expanded in terms of eigenvalues |q̂〉 of W0 we already found

VjPGVPk = VjPG0VPk + VjPG0HPQ1

E −HQQ −HQPG0HPQ + i∆/2HQPG

0VPk (70)

= VjPG0VPk +∑qj′k′

Ĉj′qVjPG0HPj′1

E − Êq + i∆/2Hk′PG

0VPkĈk′q (71)

where

VjP =√

i∆

E − E(Q)j + i∆/2

∑c

∫r2dr〈Qj |H|r, c〉〈r, c|. (72)

1We are going to use only the kinetic energy term for the scattering waves. In this case, T 0if = 0.

12

Using the definition of Mjk, eq. (56), we can rewrite this equation as

VjPGVPk =√

i∆

E − E(Q)j + i∆/2

√i∆

E − E(Q)k + i∆/2×

Mjk + ∑qj′k′

Ĉj′qMjj′1

E − Êq + i∆/2Mk′kĈk′q

(73)

Diagonalizing the denominator of Eq. (66) in terms of the eigenfunctions |q〉 of W ,

(Eq −W )|q〉 = 0, (74)

Eq. (66) can be rewritten as

Tif = Tif +∑qjk

CjqV̂ij1

E − EqCkqV̂kf , (75)

where we define

V̂ij ≡ 〈Ψi|PV|Qj〉 =∑c

∫r2drΨic(r)〈r, c|V|Qj〉 =

√i∆

E − E(Q)j + i∆/2

∑c

∫r2drΨic(r)Hcj(r) (76)

and where we used the fact that |q〉’s are linear combinations of |Qj〉

|q〉 =∑j

Cjq|Qj〉. (77)

It is then straight-forward to recognize that, in Eq. (46),

giq =√

2π∑j

CjqV̂ij . (78)

The channels c and c′ in eq. (46) denote the different channels for the scattering matrix Scc′ . Wecan perform the energy average of the coupling term in eq. (46) as〈∑

q

gqcgqc′

E − Eq

〉∆

' I2π

∫ E+n∆E−n∆

dE′∑

q gqc(E′)gqc′(E

′)/(E′ − Eq)(E − E′)2 + ∆2/4

=I∆

2π

NE∑n=−NE

∑q gqc(E + n∆)gqc′(E + n∆)

(E + n∆− Eq)[(n∆)2 + ∆2/4

] , (79)where NE is the number of energy grid points in the continuum, and ∆ = n∆/NE . The integer ndefines the integration limits. This average is therefore dependent on 3 parameters: ∆, NE , and n.

To define the Q-space we need the real energies EQj ≡ Ej . The average spacing between thecontinuum levels, Ej , will be called D, so that D = 〈Ej+1 − Ej〉. We will also need an energy rangefor the space, which we take as E −NQ∆ ≤ E(Q)j ≤ E +NQ∆. The numbers of the energy levels will

13

be denoted by NL. Thus, in order to define the Q-space, we need to define the E(Q)j ’s, NL, and NQ

(see Ref. [8]).In eq. (60) the interaction is in our model space given by Hij(r) =

∑l hijlδ (r − rl) /rlr, where i

and j denote the energies (channels) Ec ≡ Ei = E−E∗c , and Ej in P and Q space, respectively. E∗c isthe excitation energy. The index l denotes the position on the coordinate space where the interactionis active (the discrete points rl). With that, eq. (65) becomes

Ĥij =∑c

∫r2drφ∗ic(r)Hcj(r) =

Nc∑c=1

NR∑l=1

φ∗ic(rl)hcjl, (80)

where Nc is the number of open channels, and NR is the number of the rl points in the coordinate mesh.Another parameter is needed to account for the size of the coordinate region where the interaction isactive. We call this R which has the value of a typical nuclear radius. There will be NR points atpositions rl within 0 ≤ rl ≤ R. Thus, to calculate the integrals we need the additional parameters Rand the NR values of rl.

Similarly, the matrix element in eq. (54) is written as

Mjk (Ec) = HjPG0HPk =

Nc∑c,c′=1

NR∑l=1

NR∑l′=1

hcjlhc′kl′G0cc′(rl, rl′). (81)

The free particle Green’s function in the s-wave channel is given by

G0cc′(E; r, r′) ≡ G0(Ec, r, r′)δcc′ = · · · , (82)

where r< (r>) is the smaller (larger) of (r,r′). We can rewrite eq. (81) as

Mjk (Ei) =[HjPG

0HPk]

(Ei) = · · · . (83)

This determines the matrix Wjk in eq. (53).

At this point, it is worthwhile to rewrite eq. (76) by using the definition of matrices Mjk and Ĥij ,eqs. (56) and (65), respectively. In these terms

V̂ij(E) =√

i∆

E − E(Q)j + i∆/2

{Ĥik+

∑kj′q

ĈjqĤij1

E − Êq + i∆/2MkjĈkq

. (84)Thus, V̂ij can be easily calculated after we obtain matrices Mjk and Ĥij , eqs. (56) and (??),

respectively. The same applies to VjP .A numerical test of the optical background formalism to treat a large number of channels statis-

tically was done in Ref. [9] with 400 equidistant q-levels, 40 channels, with 20 equidistant coordinatepoints where HPQ is set to a Gaussian-distributed random interaction. The energy of the single-outP state was taken as E = 20 MeV, and we included 100 E′ points for Lorentzian averaging between18 and 22 MeV. The adopted value of ∆ = 0.5 MeV and we considered s-wave scattering only, withΓ/D � 1. Figures 3 and 4 show the avert of the fluctuating part of the T-matrix and the optical

14

T-matrix. It is evident that for a large number of weakly coupled channels the optical backgroundformalism yields a proper treatment of the S- and T-matrices. The advantage of the formalism is thatone has only to consider the strongly coupled P-space states, with a simple average needed for theweakly coupled states which can be numerous.

5 Conclusions

CDDC is an important tool to describe reactions with weakly bound nuclei, for which numerous statesin the continuum (resonant and non-resonant) have an important weight for the reaction cross section.A drawback of the formalism is that if the number of states involved (channels) are too large, thecalculations converge too slowly or might not be feasible with present computer resources. A possibletreatment of a large number of weakly bound states is the use of statistical methods. In this work wehave proposed several possibilities to include the average over continuum couplings: (a) perform anaverage of the coupled-channels equations where continuum-continum couplings are treated by meansof statistical ensembles, (b) solve time dependent balance equations with statistical averages, (c) usethe optical background formalism based on the Feshbach projection method, or (d) a combination ofall methods discussed here. Several applications in quantum optics, atomic and nuclear physics arepossible, specially in the area of open quantum or mesoscopic systems.

We acknowledge beneficial discussions with G. Arbanas and A. Kerman. This work was partiallysupported by the US-DOE grants DE-SC0004971 and DE-FG02- 08ER41533.

References

[1] C.A. Bertulani and L.F. Canto, Nucl. Phys. A 539 163 (1992).

[2] P. Descouvemont and M. S. Hussein, Phys. Rev. Lett. 111, 082701 (2013).

[3] M. Kawai, A. K. Kerman, K. W. McVoy, Ann. Phys. (NY) 75, 156 (1973).

[4] D. Agassi, C. M. Ko, H. A. Weidenmüller, Ann. Phys. (NY) 107, 140 (1977).

[5] C. M. Ko, Z. Phys. A 286, 405 (1978).

[6] A. K. Kerman and K. W. McVoy, Ann. Phys. (NY) 122, 197 (1979).

[7] C.A Bertulani and G. Baur, Phys. Rep. 165, 185 (1988).

[8] G. Arbanas, C.A. Bertulani, D.J. Dean, A.K. Kerman, and K.J. Roche, Eur. Phys. J. Conf. 21,07002 (2012).

[9] G. Arbanas and C.A. Bertulani, Oak Ridge National Laboratory, unpublished (2007).

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1 Statistical Continuum Discretized Coupled Channels2 CDCCS in the random matrix formalism3 Time-dependent CDCCS4 Optical background CDCCS5 Conclusions